Question

On Friday, Kaitlyn went to the park to work out. She started with a slow jog at a constant rate for five minutes. She then increased her speed to a faster, constant pace for seven minutes. She took a three-minute break, then ran for another 7.5 minutes at a constant pace to finish her two-mile run. She rested for the remainder of the time. During what domain interval will she reach 1.25 miles?


12≤t≤15

15≤t≤22.5

5≤t≤12

22.5≤t≤25

Answer 1

15 ≤ t ≤ 22.5

Let s be her slow pace (mi/min) and f her faster pace (for both the 7-min and 7.5-min runs). Then
5s + 14.5f = 2.
If she had already reached 1.25 miles by t = 12, then 5s + 7f ≥ 1.25, so 7.5f ≤ 0.75 and f ≤ 0.1.
But then 5s = 2 − 14.5f ≥ 2 − 1.45 = 0.55, so s ≥ 0.11 > f, contradicting that s < f.
Thus she cannot reach 1.25 by t = 12 (or during the break), so she must reach it during the final run: 15 ≤ t ≤ 22.5.